Inverse Dirichlet distribution

Last updated April 09, 2019

In statistics, the inverse Dirichlet distribution is a derivation of the matrix variate Dirichlet distribution. It is related to the inverse Wishart distribution.

Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

In statistics, the matrix variate Dirichlet distribution is a generalization of the matrix variate beta distribution.

Suppose $U_{1},\ldots ,U_{r}$ are $p\times p$ positive definite matrices with a matrix variate Dirichlet distribution, $\left(U_{1},\ldots ,U_{r}\right)\sim D_{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)$ . Then $X_{i}={U_{i}}^{-1},i=1,\ldots ,r$ have an inverse Dirichlet distribution, written $\left(X_{1},\ldots ,X_{r}\right)\sim \operatorname {ID} \left(a_{1},\ldots ,a_{r};a_{r+1}\right)$ . Their joint probability density function is given by

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

\left\{\beta _{p}\left(a_{1},\ldots ,a_{r};a_{r+1}\right)\right\}^{-1}\prod _{i=1}^{r}\det \left(X_{i}\right)^{-a_{i}-(p+1)/2}\det \left(I_{p}-\sum _{i=1}^{r}{X_{i}}^{-1}\right)^{a_{r+1}-(p+1)/2}

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Dirichlet distribution probability distribution

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References

A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.

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